Báo cáo toán học: "Pfaffian orientation and enumeration of perfect matchings for some Cartesian products of graphs" doc

Abstract The importance of Pfaffian orientations stems from the fact that if a graph G is Pfaffian, then the number of perfect matchings of G as well as other related problems can be co

Pfa ffian orientation and enumeration of perfect

Feng-Gen Lin and Lian-Zhu Zhang† School of Mathematical Sciences, Xiamen University

Xiamen 361005, P.R.China E-mail: zhanglz@xmu.edu.cn Submitted: Dec 13, 2008; Accepted: Apr 14, 2009; Published: Apr 30, 2009

Mathematics Subject Classification(2000): 05A15; 05C70

Abstract

The importance of Pfaffian orientations stems from the fact that if a graph G is Pfaffian, then the number of perfect matchings of G (as well as other related problems) can be

com-puted in polynomial time Although there are many equivalent conditions for the existence

of a Pfaffian orientation of a graph, this property is not well-characterized The problem is that no polynomial algorithm is known for checking whether or not a given orientation of a graph is Pfaffian Similarly, we do not know whether this property of an undirected graph that it has a Pfaffian orientation is in NP It is well known that the enumeration problem

of perfect matchings for general graphs is NP-hard L Lov´asz pointed out that it makes sense not only to seek good upper and lower bounds of the number of perfect matchings for general graphs, but also to seek special classes for which the problem can be solved

exactly For a simple graph G and a cycle C n with n vertices (or a path P n with n vertices),

we define C n (or P n ) ×G as the Cartesian product of graphs C n (or P n ) and G In the present paper, we construct Pfaffian orientations of graphs C4 × G, P4× G and P3× G, where G

is a non bipartite graph with a unique cycle, and obtain the explicit formulas in terms of eigenvalues of the skew adjacency matrix of→−G to enumerate their perfect matchings by

Pfaffian approach, where→−G is an arbitrary orientation of G.

1 Introduction

The theory of Pfaffian orientations of graphs had been introduced by the physicists M E Fisher, P W Kasteleyn, and H N V Temperley The importance of Pfaffian orientations stems

∗ This work is supposed by NFSC (NO.10831001).

† Corresponding author.

from the fact that if a graph G is Pfaffian, then the number of perfect matchings of G (as well as

other related problems) can be computed in polynomial time

P W Kasteleyn gave a polynomial time algorithm for computing the number of perfect matchings in planar graphs using Pfaffian method and extended his approach to toroidal grids

in [6] and [8] Litte [9] generalized P W Kasteleyn’s work and proved that if a bipartite graph

Little [3] proved that a graph has a Pfaffian orientation under which every cycle of even length

is clockwise odd if and only if the graph contains no subgraph which is, after the contraction

of at most one cycle of odd length, an even subdivision of K2,3 McCuaig [11], and McCuaig, Robertson et al [12], and Robertson, Seymour et al [15] found a polynomial-time algorithm to determine whether a bipartite graph has a Pfaffian orientation respectively In spite of there are many equivalent conditions for the existence of a Pfaffian orientation of a graph, this property

is not well-characterized The problem is that no polynomial algorithm is known for checking

whether or not a given orientation of a graph G is Pfaffian We do not even know whether

this property is in NP (It is trivially in co-NP; to prove that a given orientation is non-Pfaffian,

it suffices to exhibit two perfect matchings with different signs.) similarly, we do not know whether the property of an undirected graph that it has a Pfaffian orientation is in NP

The number of perfect matchings is an important topological index which has been ap-plied for estimation of the resonant energy and total π−electron energy and calculation of paul-ing bond order (see [4], [13], [16]) Enumeration problem for perfect matchpaul-ings in general graphs(even in bipartite graph) is NP-hard L Lov´asz [10] pointed out that it makes sense not only to seek good upper and lower bounds of the number of perfect matchings for general graphs, but also to seek special classes for which the problem can be solved exactly So far, many mathematicians, physicists and chemists have focused most of their attention on the enu-meration problem for perfect matchings (see [2], [3], [5], [14], [15])

First, we repeat some standard definitions A graph G is a pair V(G) and E(G), where V(G)

is a finite set of vertices and E(G) is a set of unordered pairs xy of vertices called edges We say that the edge xy is incident with x, y and that x, y are adjacent and are called the ends of the edge

xy All graphs in this paper are simple graphs which are finite, do not have loops or multiple

edges A graph H is a subgraph of G if V(H) ⊆ V(G) and E(H) ⊆ E(G) A spanning subgraph

of G is a subgraph H with V(H) = V(G) For a nonempty subset V′ of V(G), a subgraph of G

is called induced subgraph induced by V′if its vertex set is V′ and edge set is the set of those

edges of G that have both ends in V′ A k-path denoted by x0x1 .x k is the graph with distinct

vertices x0,x1, ,x k and edges x i−1 x i , i = 1, 2, , k, where x0,x k are called its ends A k-cycle

is obtained from a (k − 1)-path by adding the edge between the two ends x0,x k−1 We say that

and C n respectively A graph is connected if any two vertices are joined by a path A bipartite

graph is one whose vertex set can be partitioned into two disjoint subsets X and Y, so that each edge has one end in X and one end in Y; such a partition (X, Y) is called a bipartition of the

graph A graph is bipartite if and only if each cycle of it has even length A tree is a connected

acyclic graph Clearly, a tree is bipartite A perfect matching of a simple graph G is a set of vertex-disjoint edges that are collectively incident to all vertices A cycle C of G is said to be

nice if G − C contains a perfect matching, where G − C denotes the induced subgraph of G obtained from G by deleting the vertices of C If C is a nice cycle of a spanning subgraph of G, then C is also a nice cycle of G Let→−

G denote an orientation of G which is obtained from G by

specifying, for each edge, an orientation on its ends If C is an even undirected cycle in→−G, we

shall say C is evenly oriented if it has an even number of edges oriented in the direction of the routing Otherwise C is oddly oriented An orientation→−

G is Pfaffian if every nice cycle of G is

oddly oriented in→−

G A graph G is Pfaffian if it has a Pfaffian orientation.

For V(G) = {v1,v2, ,v n}, the skew adjacency matrix of→−G, denoted by A(→−

G), is defined as

follows:

A(→−











1 if (v i,v j ) ∈ E(→−G),

−1 if (v j,v i ) ∈ E(→−G),

0 otherwise

Let←−

The cartesian product of two graphs G and H denoted by G × H is the graph with vertex set

V(G) × V(H) such that (x, u) and (y, v) are adjacent iff either x = y and u and v are adjacent in

H, or u = v and x and y are adjacent in G For bipartite graphs, Yan and Zhang in [18] studied

the enumeration of perfect matchings for these Cartesian product of graphs C4 × T , P4 × T

and P3× T , where T is a tree In the present paper, we construct Pfaffian orientations of some

Cartesian product of graphs which are non-bipartite and obtain explicit formulas to enumerate their perfect matchings by Pfaffian approach as follows

(1) φ(C4× G) = Q

λ ∈λ(→−G )

(2 − λ2); (2) φ(P4× G) = Q

λ ∈λ ∗ ( → −G )(1 − 3λ2+ λ4);

(3) If G has a perfect matching, then φ(P3× G) = Q

λ ∈λ ∗ ( → −G )(2 − λ2) and φ(C4× G) = φ(P3× G)2,

where G is a non-bipartite graph with a unique cycle,→−G is an arbitrary orientation of G, λ(→−G)

is the set of all eigenvalues of A(→−

G) is the set of those non-negative imaginary part

eigenvalues of A(→−

G).

2 Pfa ffian orientation

G be an ori-entation of G Then the following three properties are equivalent:

G is a Pfaffian orientation.

G.

(3) If G has a perfect matching, then for some perfect matching F, every F-alternating cycle is

G.

For a simple graph G with V(G) = {v1,v2, ,v n }, let G1 and G2 be two copies of G with V(G1) = {v′

1,v′

2, ,v′

n } and V(G2) = {v′′

1,v′′

2, ,v′′

n } respectively, where v′

i in G1 and v′′

i

in G2 are corresponding to v i in G (i = 1, 2, · · · , n) Adding the edges v′i v′′i (i = 1, 2, · · · , n) between G1 and G2, the resulting graph is P2× G with vertex set V(G1) ∪ V(G2) and edge set

E(G1) ∪ E(G2) ∪ {v′

i v′′

i | i = 1, 2, · · · , n} If→−G is an orientation of G, then we denote the

orien-tation of P2× G by (P2 ×→−G) e which is defined as follows: the orientation of G1 (the left half

of P2× G) is the same as→−G and that of G2 (the right half of P2× G) is the same as←G, and the−

orientations of edges v′

i v′′

i (i = 1, 2, · · · , n) are from v′

i (see Figure 1)

2

Figure 1.

G is an orientation of G under which every cycle

is a nice cycle and can be written as:

v′i

1v′i

2 .v′i

m v′′i

m v′′i

m−1 .v′′i

2v′′i

1v′i

where i1,i2, ,i m ∈ {1, 2, , n}.

i −v′′

j −v′′

i for

i −v′′j and

P v′

j −v′′

i for i , j in P2× G, then there is a cycle of P2× G that consists of P v′

i −v′′

j , v′′

j v′

j , P v′

j −v′′

i and

v′′

i v′

i, which has the following form:

v′i .v′′j v′j .v′′i v′i (2)

In the other hand, by Lemma 2.3, every cycle of P2×G has the form (1) in Lemma 2.3 It is clear

that the cycle form (2) is distinct from the cycle form (1), a contradiction The assertion holds ❏

In order to formulate our main results, it is necessary to introduce further terminology

Suppose G is a non-bipartite graph with a unique cycle For convenience, let C∗denote the

unique odd cycle of G with length 2k + 1, and label the vertices of G as v1,v2, ,v 2k+1, ,v n

such that C∗ = v1v2 .v 2k+1 v1(see Figure 2(a)) In P2 × G(see Figure 2(b)), let the cycles C∗

i

in G i (i=1, 2) be corresponding to the cycle C∗in G For E′ ⊆ E(G), G − E′ denotes the graph

obtained from G by deleting the edges in E′ If E′ = {e} we write G − e instead of G − {e} A

path with ends s and t is denoted by P s−t

v2

v3

v4

v5

v2

k-v2 -1k

v 2k

v2 +1k

1

v2 +1k

v2 v

3

v4

v5

v2

k-v2 -1k

v 2k

1

v2

v3

v4

v5

v2

k-v2 -1k

v 2k

v2 +1k

1

v2

v3

v4

v5

v2

k-v2 -1k

v 2k

v2 +1k

1

v2 +1k

v2 v

3

v4

v5

v2

k-v2 -1k

v 2k

2 1 2 +1k ) ( ) a G

Figure 2.

is a nice cycle of it.

in P2 × G, and E′ = {v′

1v′

2k+1,v′′

1v′′

2k+1} If eC contains no edge of E′, then eC is an even cycle of

P2× (G − v1v 2k+1 ) (see Figure 2(c)) which is a spanning subgraph of P2× G Since G − v1v 2k+1

is a tree, eC is a nice cycle of P2× (G − v1v 2k+1) by Lemma 2.3 Thus eC is a nice cycle of P2× G.

the edge v′

1v′

2k+1 Since G − v1v 2k+1 is a tree, it is bipartite If (V1,V2) is its a bipartition, then

P2×(G −v1v 2k+1 ) is a bipartite graph with a bipartition (V′

1∪V′′

2,V′

2∪V′′

1), where both V′

i and V′′

i

are corresponding to V i , i = 1, 2 Furthermore, there exists a 2k-path v′1v′2 .v′i v′i+1 .v′2k v′2k+1,

so the two vertices v′1, v′2k+1 belong to the same partitioned subset Hence eC − v′1v′2k+1 is a path

P v′

1−v′2k+1 of P2 × (G − v1v 2k+1) which always has even length Thus the length of eC is odd, a

contradiction

1−v′′1 and

P v′

2k+1 −v′′

2k+1 of P2 × (G − v1v 2k+1 ) Otherwise, there are two disjoint paths P v′

1−v′′

2k+1 and P v′

2k+1 −v′′ 1

which contradicts to Lemma 2.4 By application of Lemma 2.3, we have P v′

1−v′′

1 = v′

1v′′

1 or

v′1v′i

1 .v′i

s v′′i

s .v′′i

1v′′1, and P v′

2k+1 −v′′

2k+1 = v′2k+1 v′′2K+1 or v′2k+1 v′j

1 .v′j

t v′′i

t v′′j

1v′′2k+1 respectively

Thus, {v′l v′′l | 1 6 l 6 n, v′l <V( e C)} is a perfect matching of G − e C Therefore e C is a nice cycle.❏

By Theorem 2.5, the following corollary is immediate

Suppose G is a non-bipartite graph with a unique cycle and→−

G is an arbitrary orientation of

cycle in (P2×→−G) e is oddly oriented by Theorem 2.1 By Theorem 2.5, the orientation (P2×→−G) e

is an orientation of P2× G under which every even cycle of P2× G is oddly oriented Now we

apply Lemma 2.2 with G replaced by P2× G, then (P2× (P2×→−G) e)e is a Pfaffian orientation of

P2×(P2×G) Since P2× P2 = C4, we use (C4×→−G) e instead of (P2×(P2×→−G) e)e for convenience Figure 3 illustrates the orientation procedure

( ) ( a P G

2 ) ( ) (b C4 G)

Figure 3.

For G with V(G) = {v1,v2, ,v n }, take m copies of G, denoted by G i with V(G i) =

{v (i)1 ,v (i)2 , ,v (i) n }, i = 1, 2, , m P m × G is the graph with vertex set

m

S

j=1

V(G i) and edge set

m

S

j=1

E(G i ) ∪ {v (i) j v (i+1) j |1 6 j 6 n, 1 6 i 6 m − 1} Let→−G be an orientation of G We define the

orientation of G i in P m × G to be the same as→−G if i is odd,←−

G otherwise, and the orientations of

edges v (i) j v (i+1) j in P m × G to be from v (i) j to v (i+1) j (1 6 j 6 n, 1 6 i 6 m − 1) The orientation of

P m × G defined as above is denoted by (P m×→−G) e The processes of the orientations (P3×→−G) e

and (P4×→−G) eare shown in Figure 4

Figure 4.

Since P4×G is a spanning subgraph of C4×G, every nice cycle in P4×G is also a nice cycle

in C4× G Noting that (P4×→−G) e is the orientation (C4×→−G) e restricted in P4× G and (C4×→−G) e

is a Pfaffian orientation, we obtain that every nice cycle in P4× G is oddly oriented relative to

(P4×→−G) e Then we get the following theorem immediately

G be an arbitrary

orientations.

G be an arbitrary

perfect matching of G4in P4× G Clearly M1∪ M2is a perfect matching of P4× G − C So that

every nice cycle in P3× G is also a nice cycle in P4× G Moreover, (P3×→−G) eis the orientation

(P4×→−G) e restricted in P3× G, and (P4×→−G) eis a Pfaffian orientation by Theorem 2.7, so every

nice cycle in P3 × G is oddly oriented relative to (P3×→−G) e By Theorem 2.1, (P3×→−G) e is a

3 Enumeration of perfect matchings

If a graph G has a Pfaffian orientation→−G, then the number of perfect matchings of G denoted

by φ(G) can be computed in polynomial time by the following theorem.

G be a Pfaffian orientation of a graph G Then

φ(G)2= det A(→−G),

G be an arbitrary orientation of G Then

φ(C4× G) = Y

λ ∈λ(→−G )

(2 − λ2),

G).

of (C4×→−G) e has the following form by a suitable labeling of vertices of (C4×→−G) e:













−I −A(→−G) 0 I

−I 0 −A(→−G) −I

G)













= C D A B

! ,

where I is the identity matrix, A =





 A(

−

→

−I −A(→−G)





, B = 0 I I 0

!

, C = −I0 0

−I

!

, D =





 −A(

−

→







It is well known that for four matrixes A, B, C, D with equivalent order n, if det A , 0 and

!

= det(AD − CB) By Theorem 3.1, we have

φ(C4× G)2 = det A((C4×→−G) e)

= det





−





 A(

−

→

−I −A(→−G)







2

+ 0I 0I !



= det





 2I − (A(

−

→

0 2I − (A(→−G))2







= (det(2I − A(→−G)2))2

Since A(→−

G) is a real skew matrix, its eigenvalues are either zeros or pure imaginary numbers,

φ(C4× G) =

det(2I − A(→−G)2)

= Y

λ ∈λ(→−G)

(2 − λ2),

where λ(→−

Corollary 3.3 Let G be an odd cycle with 2k + 1 vertices Then

φ(C4× G) =

2k+1

Y

j=1

2 + 4 sin2 2 jπ

2k + 1

!!

Proof Without loss of generality, we orient every edge of the odd cycle G clockwise Then

the skew adjacency matrix A(→−G) is a circulant matrix [1], and the eigenvalues of A(→−G) are

λj = 2i sin( 2k+1 2 jπ ), j = 1, 2, , 2k, 2k + 1 By Theorem 3.2, the assertion holds. ❏

In this case, the author of paper [6] had presented a rigorous but more complex solution to enumerate its perfect matchings

G be an arbitrary orientation of G Then

φ(P4× G) = Y

λ ∈λ ∗ ( → −G)

(1 − 3λ2+ λ4),

we have

φ(P4× G)2= det A((P4×→−G) e)

By a suitable labeling of vertices of (P4×→−G) e , the skew adjacency matrix of (P4×→−G) e has the following form:













−I −A(→−G) I 0













,

Now multiplying the first column, then the third and fourth row, then the fourth column of

the partitioned matrix A((P4×→−G) e) by −1, without changing the absolute value of the determi-nant we obtain the matrix

M =













−A(→−G) I 0 0













where ⊗ denotes the Kronecker product of matrices and

B =











0 1 0 0

1 0 1 0

0 1 0 1

0 0 1 0









.

It is well known the eigenvalues of −I4⊗ A + B ⊗ I nare

µi − λj (1 6 i 6 4, 1 6 j 6 n),

where λ1, λ2, , λn are the eigenvalues of A(→−

It is easy to calculate that the eigenvalues of B are

±

s

3 + √5

s

3 − √5

Thus the eigenvalues of M are

±

s

3 + √5

2 − λs,±

s

3 − √5

2 − λs,(s = 1, 2, , n).

Since the determinant of the matrix M is the product of these eigenvalues,

det A((P4×→−G) e)
= |M|

=

n

Y

s=1

q

3+ √ 5

2 − λs

!

−

q

3+ √ 5

2 − λs

! q

3−√5

2 − λs

!

−

q

3−√5

2 − λs

!

=

n

Y

s=1

(1 − 3λ2s+ λ4s)

If λ is an eigenvalue of the real skew matrix A(→−

G), so is its conjugate λ Hence we have

φ(P4× G) =

q

det(A((P4×→−G) e))

=

n

Y

s=1

q

(1 − 3λ2

s)

= Y

λ ∈λ ∗ ( → −G )

(1 − 3λ2+ λ4),

where λ∗(→−

G) The Theorem

Similarly, by using Theorem 2.8, we can prove the following Theorem.

orientation of G If G has a perfect matching, then

φ(P3× G) = Y

λ ∈λ ∗ ( → −G)

(2 − λ2),

G be an arbitrary

Acknowledgements

We wish to thank Professor Weigen Yan for too much useful help and advice

References

[1] N Biggs, Algebraic Graph Theory, Cambridge, Cambridge University Press, 1993 [2] M Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J Com-bin Theory Ser A 77(1997), 67–97

[3] Fischer, C.H.C Little, Even Circuits of Prescribed Clockwise Parity, Electron J Combin

10 (2003) #R45

[4] G G Hall, A Graphic Model of a Class of Molecules, Int J Math Edu Sci Technol., 4(1973), 233–240

[5] W.Jockusch, Perfect matchings and perfect squares, J Combin Theory Ser A 67(1994), 100–115

[6] P W Kasteleyn, The statistics of dimers on a lattice, Physica., 12(1961), 1209–1225 [7] P W Kasteleyn, Dimer statistics and phase transition, J Math Phys 4(1963), 287–293 [8] P W Kasteleyn, Graph Theory and Crystal Physics In F.Harary, editor, Graph Theory and Theoretical Physics Academic Press, 1967, 43–110

[9] C.H.C.Little, An extension of Kasteleyn’s method of enumerating the 1-factors of pla-nar graphs, in: D.Holton, ed., Combinatorial Mathematics, Proceedings 2nd Australian Conference, Lecture Notes in Mathematics 403 (Springer,Berlin, 1974) 63–72

[10] L Lov´asz and M Plummer, Matching Theory, Ann of Discrete Math 29, North-Holland, New York, 1986

[11] W McCuaig, P´olya’s permanent problem, Electron J Combin 11 (2004), #R79

[12] W McCuaig, N Robertson, P D Seymour, and R Thomas, Permanents, Pfaffian ori-entations, and even directed circuits (Extended abstract), Proc 1997 Symposium on the Theory of Computing (STOC)

[4] G G Hall, A Graphic Model of a Class of Molecules, Int J Math Edu Sci Technol., 4(1973), 233–240

[5] W.Jockusch, Perfect matchings and perfect squares, J Combin Theory Ser A...

In this case, the author of paper [6] had presented a rigorous but more complex solution to enumerate its perfect matchings

G be an arbitrary orientation of G Then

φ(P4×... Professor Weigen Yan for too much useful help and advice

References

[1] N Biggs, Algebraic Graph Theory, Cambridge, Cambridge University Press, 1993 [2] M Ciucu, Enumeration